^ ARE No. L4L14 NATIONAL ADVISORy COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED January 19i«-5 as Advaxice Restricted Report L4L1U CRITICAL COhEIHATIONS OF SHEAR AMD TRANSVERSE DIRECT STRESS FOR AN INFINITELy LONG FLAT PLATE WITH UXZS ELASTICALLY RESTRAINED AGAINST ROTATION By S. B. Eatdorf and John C. Houl)olt Langley Memorial Aeronautical Laboratory Langley Field, Ya. UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 1 20 MARSTON SCIENCE LIBRARY NACA ■L 32611-7011 USA WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre- viously held under a security status but are now unclassified. Some of these reports were not tech- nically edited. All have been reproduced without change in order to expedite general distribution. L - 183 ]IH ^n 'Y^H ^rocff^r NACA ARR No. L4L14 NATIONAL ADVISORY CO?;ffi'ITTEE FOR AERONAUTICS ADVANCE RESTRICTED xRSPORT CRITICAL COMBINATIONS OF SHEAR AND TRANSVERSE DIRECT STRESS FOR AN INFINITELY LONG FLAT PL/.TE WITH EDGES ELaSTICALLY RESTRAINED AGAINST ROTATION Py S. B. Batdorf and John C. lioubolt SUT-1MARY An exact solution and a closely concurring approxi- mate energy solution are given for the buckling of an infinitely long flat plate under combined shear and transverse direct stress vn th edges elastically restrained against rotation. It was found that an appreciable frac- tion of the critical stress in pure shear may be applied to the plate vithout any reduction in the transverse compressive stress necessary to produce buckling. An interaction formula in general use vvas shov;n to be decidedly conservative for the range in which it is sup- posed to appl;/. INTRODUCTION In the design of stressed-skln structures, considera- tion must sometimes be given to the critical stresses for a sheet under a combination of shear and direct stress. The upper surface of a wing in normal flight, for example, is subjected to combined shear and compressive stress, and the lower surface is subjected to combined shear and tensile stress. The upper surface maj?- then buckle at a lower compressive or shear stress than if either stress were acting alone. The critical shear stress for the lower surface may be increased by the presence cf the tensile stress. NACA ARE No. L4L14 Tx the v/ing ha the skin between tw as a long sheet ?li chordvi/ise directior: span'.vise direction stress. A conserva critical stresses ni sidered to be flat paper, the ci-itical itely long flat -pie The corre spondins i v/ith spanwise stiff treated in referer.c s closely spaced choroA'ice stlffeners, o adjacent stlffeners may be regarded ghtly curved in the longitudinal or and straight in tlie transverse or loaaed in shear and transverse direct tive preliminary estimate of the ay be obtained, if the sheet is con- and infinitely long. In the present stresses are computed for an infin- te loaded as indicated in figure 1(a). dealizaticn of the case of a wing eners, shown in figure 1(b), was e ± . (a) i. .L.U. l^l^L i jua <-"> ,.-">J> 'ype of loadjng problem solved in present paper, 1 -^ jl \l C_' ':s^ I .J L (b) Type of loading problem solved in reference 1, Figure 1.- Buckling of an infinitely long plate under combined loads. NACA ARR No. L4L14 3 ccnventtc:tal interaction forctlas For buckling of structures under combined loading conditions, no general theory has been developed that is applicable to all ca?es. Stress ratios (reference 2) hovvever, provide a convenient 'iiethod of representing such conditions. For example, the ratio of the shear stress actually present in a structure to the critical shear stress of the structure when no other stresses are present may be called the shear-stress ratio. Stress ratios iray similarly be defined for each type of stress occurring in the structure. It is generally assumed that equations of the type may be used to ezpress the buci-cllne conditions in the case of combined loading (reference 3, p. 1 - 18). In equation (1), R-^ , R2, and R^^^ are stress ratios and p, q, and r are exponents chosen to fit the known results. (All symbols are defined in appendix A.) Such a formula gives the correct resu]-ts when only one type of loading' is present and has the further advantage of being nondimenslonal . Equation (1) implies, m.ore- over, that the presence of any positive fraction of the critical stres? of one type reduces the amount of another type of stress required to produce buckles; this Implication appears reasonable and has been proved true in some cases (references 1, 2, and 4). In reference 2 the following interaction formula is given for an inl'initely lonp plate with clamped edges loaded in shear and longitudinal compression: R.^"^ + Re = 1 (2) where Rq is shear- stress ratio and R^ is longitudinal direct-stress ratio. The same formiula is recomniended in I'eference 3 for general use for the buckling of any flat 4 NAG A ARR No. L1L14 rectangular plate, regardless of the direction of com- pression and the degree of edge restraint. Later theoretical work (reference 1) shows that, to a high degree of accuracy, for an infinitely long plate V'ith any degree of edge restraint loaded in shear and longitudinal compression Rg^ + R^ = 1 (3) The sar_e formula was found in reference 5 to he appli- cable to simply supported rectangular plates of aspect ratios 0.5, 1, and 2; and the conclusion wa^; drawn that interaction curve? in stres--ratio form are practically independent of the dimensions of the plate. The present analysis, however, indicates that the buckling of an infinitely long plate loaded in shear and transverse compression (fig. 1(a)) is not adequately represented either by equation (2) or (3) or by any formula of the type of equation (1). Two independent theoretical solutions to this buckling problem, are given in appendixes 3 and G. Appendix E contains the exact solut5.on of the differential equation of equilibrium, and appendix C contains an energy solution leading directly to an interaction form.ula. This energy solu- tion, which gives approximate values only, v/as miade to obtain an initial quick survey of the problem, and to provide a check on the results of the exact solution. Approximate interaction formiulas in substantial agreement with these results were given for the cases of sim.ply supported and clam.ped edges in reference 6. RESULTS AIID DISCUSSION In figure 2, curves are given that indicate the critical combinations of shear and transverse direct stress for an infinitely long plate with edges elasti- cally restrained against rotation. These curves are computed from the exact solution presented in appendix B. The degree of edge restraint is denoted by f, which is defined in appendix B in such a way that zero edge restraint corresponds to simply supported edges and NACA ARR No. I:4L14 Infinite edge restraint indicate.^ clarped edges. A similar set of curve- is given in tonus of stz'-ess ratios in figure 2. The nur.ierica] values uped to plot figures 2 and 5, together Vv'ith the values fo^ond by the energy solution, are given in table I, The mosb striking feature of these resultr is that an appreciable fraction of the critical stress in pure shear can evid'^ntly be applied to the plate without any reduction in the conpresiive stress nece.'^sary to produce buckling. (Sec fig. Z .) Thi.-: fraction varies from about one- third to more than one-half, depending on the degree of restraint. At shear stresses higher than those corresponding to this fraction, the ccinpressive stress required to produce buckling Is reduced bj the presence of shear. The result that the compressive buckling stress is entirely unaffected by the presence of a con- siderable amount of shear is probably peculiar to infin- itely long plates. It is to oe expected, however, that this result will be closely approached in the case of long finite plates. In figure 4 a corrparison is "lade between the exact solutions and the interaction formulas of equations (2) and (3), Equation (2), which is the interaction formula In general use, is seen to be decidedly conservative. C0:iCLlI3I0NS The exact solution of the differential equation for the bucklr'ng of en infinitely long flat plate ^onder comi- blned shear and transverse direct stress with edges elastically restrained against rotation indicates the following: 1, ;ln infinitely long flat plate miay be loaded with an appreciable fraction of its critical stress in pure shear without causing any reduction in the transverse compressive stress necessary to pr:iduce buckling. 2. An Interaction form.ula in general use for rec- tangular plates in combined shear and compression is FACA ARR No. L4L14 decidedly corservf tlve when applied to an infinitely long plate in shear and transverse co.ripression. Langley Memorial Aeronautical Laboratory National Advisory Com'.riittee for Aeronautics Langlev Field, Va. NACA ARR No. L4L14 APPSrLIX A SYI.IBCLS E ^V *X7 Tc V- Vc CjjCg' functions of edge restraint coefficient £ given in appendix C D f]exural stiffness of_plate per unit length, i "^ ■ Zt" in-lb ^12 (l - ^-)_ elastic modulus of laaterlal, psl compressive force per unit length, lb/in. shearing force per un?°.t length of plate, lb/in. rotational stiffness per inch of restraining member at edge of plate, Ib/rsdian worli done by compress:5ve force per half wave length, in-lb work done by shear force loer ha.lf wave length, in-lb strain energy in plate per half v/ave length, ixn-lb strain energy in edge restraint, per half wave length, In-lh Y function of y associated v.itn deflection of plate during buckling b width of plate, in. bn width of plate in oblnque coordinate system of reference 1, in. function of a, ;". , and "A, k(,,kc critical com.pressive and shear-stress coeffi- cients, respectively 8 NACA ARR No. L4L14 m root of a cliafacterlscj c sqr.ation of appendix B f]_,f2,f3 functions of restraint coefficient c given in pppendix C TTb thickness of plate, in. w displacement of buckled plats from original position w^ amplitude of assumed vavs form of buckle X longitudinal coordinate of plate y transverse coordinate of p]ate a,p functions of ,\, y, and k,, e nondir.ensicnal coeffjcient of ed£;;e restraint Y one of tvvo -parameters determ.inirg buckle form \ one of two parameters determining buckle form, (half v/ave length of bu.ckle^ in.) p. Pois'^on's ratio o direct stress o transverse direct s cress, pri T shear stress, psl angle betvveen buckle node and y-axis e = tan R-i,Ro,R-z stress ratios Rg shear-stress ratio Rq longitudinal direct-stress ratio cr critical (used as subscript) NACA ARR No. L4L14 APPENDi:: E SOLUTION BY DIFFERENTIAL EQ,UATION Stat ement of problem .- The exact solutioi: for the critical ptres? it which bi.ickling occur? in a ^"lat rectangular plate sn.bjected to combined phear and com- pres.^ion in its own plane may be obtained by solving the differential equation that expresses the equilibrium of the buckled plate. The plate is arsumed to be infin- itely long, and equal elastic restraints against rotation are assuined to be present along the two edges of the plate. Differential eq i iation .- r'igure 5 shows the CDordinate system used. The differential equation for equilibraum of a flat plate under shear and transverse direct stress is (from reference 7) + + 2Tt + at ^ ' 6y2 (Bl) It is convenient to write a„ and t in terms of the dimension] ess buckling coefficients means of the relations kr and 07 a V T = o b'^t c^rr-^^D (P2) Substitution of the ex;crecsions for from equations (E2) in equation (Bl) gives Oy and 10 NACA ARR No. L4L14 6x4 6 ^w dx'^dj^ ^7' cfT 2: .2^ ^ ^ 6^ ^ (33) 6x6y b^ (^t'^ Sol utlo r of ajfrprenbicl equation-! .- If the plate is Infinitely jong "n the z-directlop, all di replacement 3 must be periodic in x and the deflection purface may be taken in the form v/ = V e (B4) where Y is a function of y only and \ is the half wave lergth of the buckles iti the x-direction. Substitution of rhe expression for w from equa- tion (F4) in the di f f ei-cntial equation (B3) gives the following as the equation that determines ^^: d% dy4 TT^k. Vo' \^ / dy^ 2Tr'^lk ^x ^y x^ A solution of equation (E5) is . y where m is a root of the characteristic equation 171^ + 2 m rr^k. ^.^^{^:^.{0 .0 (B.) I /, Except for the substitution of -(p-) - fr^hc for 2(-- ) , equation (56) is idoi.tical v.ith equa- tion (A-6) of appendix A of reference 8, in which NACA ARR No. L4L14 11 equation (B5) of this appendix was solved with k^ = 0. With this change, all the results obtained in appendix A of reference 8 are applicable here. The stability criterion for combined compression and shear is therefore the same in form as that for shear alone, given by equa- tion (A-19) of reference 8, which is 2ap (4y^ - -^)(cosh 2a cos 2;3 - cos 4y) - [4y^(p^ - a^) sinh 2a sin 2p 0\ £■'-' -C?^ + a^T - (4y2 - ^ + a^) ' + e [a(4Y^ + a^ + p^) cosh 2a sin 2^ + 3(4y^ -a^ - ,3^) sinh 2a cos 2p - 4apY sin 4y| = (E7) The relation between k-^. and a, p, and y is also the same in form as that in equation (A-23) of . reference 8, namely. k, 8r( a2 +^ TT 2 i[]2 \ (B8) In the present renort, however, a and p have the following values: ^ a - \ / c + y(.3.e)^._^(-y v'— \/(fv*o)^-i(f; > (B9) J wnere - 2 X l/TTb\^ rr^ P ^c 12 NACA ARR- No. L4L14 As in reference 8, the rentraint coefficient c i? defined herein by the relat5.on Sb where S is the ratio of a sinusoldally applied moment to the resulting slnusoidally distributed rotation of the restraining element measured in' radians. Evaluation of kg corresponding to a selected value of k^.- The procedure tor evaluating I'c;, after values of kc and e have been chosen, is as follows; A value of bA 1- selected; a series of values of y are assujned;'until one is fourid that, together with the'' co.rre spending values of and 3 computed from, equa- tion (69), satisfies equation (B7) ; ko is then com- puted from equation (B8) . Another value of bA is seleQted; a new set of values of y* a> ^-i^*^ r ^^ found that satisfies the stability criterion; and a new value of k.s is computed. The entire -process is repeated until the minimum value of k^ can be found from, a plot of k^ against b/\. When e is a func- tion of b/^, e must, be reevaluated each timie a different value of b/\ is selected. The minlmu]:;i value of kg and the chosen valTie of kc, when inserted in equations (32) , give a critical com.binatlon of shear and direct stress. Evaluation of k^ when kc has value corresponding to buclcllnT ar E'uler column .- Cne critical comblnatl o n Ox shear and compression is simply kc, = and kc equals the value corresponding to buckling as an Luler column. The curves giving critical stress combinations, however, did not appear to be approaching this point as their construction progressed. It was therefore neces- sary to determine whether values of kp other than zero are crit'cal vrhen the Suler com.pressive stress is reached. The determ.ination of kg, when kc reaches the value at which the plate buckles as an Euler column requires special treatment, because k^, given by equation (E8) , becomes indeterminate when the wave length becomes infinite ap suggested by the energy NACA ARR No. L4L14 13 solution. The result that \ become? infinite when k^ takes its Euler value is readily checked from equa- tion (Cll) for the special case of e = 0; for this case p = q = r = ^ and (kc)^^ = 1. From equation (B8) it is clear that, if k^ is to remain finite when the wave length approaches infinity, either Y (a) or q2 + .32_^ Q (b) For case (a), mien e = 0, it follows from equa- tions (B9) that, to small quantities of the second order, a = i[5-i(3v^*2u^) TT \i - r where ^ J (BIO) u - Trb 2\ If the values of a and p from equations (BIO) are substituted in equation (B7) and the resulting equation is expanded, with only the loviest powers of u and y retained. P 4Tr2u^ 64 - 6Tr' (BID Substitution of the values of a and _3 from equa- tions (BIO) and (Bll) in equation (BS) gives as the final result for e = k. = ± 2v \f^ 6Tr^ 14 FACA AKR No. L4L14 Case (To) can be aralyz-ed by a sir.ilar method, but the analysis is quite complicated because terms of third order inust be retained. For e = and e = (x>, case (a) and case (b) v;ere found to lead to exactly the sanie result for A value of other than zero when kj3 takes its Euler value in.ay be found in the same manner for other values of edge restraint. For any value of the restraint coefficient €, + { IL- - 2£ - L- jcos >=s= 2 j •T\/k„ - — : — /"cos ttJ.-c^ - 1) 1/ .3 _ 7.2 ,_. jl- ^kc y ko - 6Tr \/ k^ - 2e tt\/ k., + ^^\\ >■ 3 + V:^ 4 ^'z j sin tt \(1zq (B12) X 2 : 3 2^ "■■ -f-e i cos TT u k. + ( U + 8e + 2€^ + -^-1 — Vcos Tr\/ k, - l) \ 2Tr-k,/ ■ where k^^ has the value corresponding to Fval-^r colurm buoklin.fT, at this value of e. The relationship between this Euler value of kc and the corresponding € is given by the equation (from refercxice 9) e = -TT^/k^ tan 'nI' ■rT\ yir NACA ARR No. L4L14 15 V.'hen kg reaches the Euler value, the critical shear- stress coefficient kg can therefore be either or the v£.li]3 riven by equation (E12), The conclusion that kg can also have any value taetv/een these 13n;its is plausible on the basis of the following physical con- siderations. The shear stress does no work during buckling when the stress condition is such that the p].ate buckles with an Infinite wave length. The effect of shear, furthermore, is to r-ed.uce the wave leiigth to a value of the order of the width of the plate. The wave length at the tir.-e buckling occurs is infinite, however, vi?hen the plate is either In pure compression or at the value of kg satisfying equation (E12). This fact means that, for values of kg between and that given by equation (B12) , the shear stress is not great enough to force buckljng in short waves and therefore does not assist in prodaclng buckles. In this range of shear stress the co-npressive stress necessary to produce buckling is, consequently, the Euler stress. 16 NACA ARR No. L4L14 APPENDIX C SOLUTION BY ENERGY I^PHOD FOR EDGE RESTRAINT INDEPEJIDENT OF WAVE LENGTH The critical stres? is determined on the basis of the principle that the elastic-strain energy stored in structure during buckling is equal to the work done by the applied loads during buckliiig. If the structure under consideration is an infinitely long plate under coinbined shear and edge compression with edges elasti- cally restrained against rotation, this equality may be written Tc ^ = '^1 + v< (CI) In reference 1 an energy solution was given for the type of loading shown in figure 1(b) . The deflection function used in reference 1 is also suitable for appli- cation to the solution of the type of loading shown in figure 1(a), which is the loading considered in the present paper. The values for Tg, ¥]_, and Vg may accordingly be taken directly from reference 1, but T^ must be recomputed to apply to the case of transverse com.pressive stress. The following substitutions are used to transform the energy expressions from the oblique coordinates of reference 1 to the rectangular coordinates used in the present paper (fig. 6): Reference 1 y/bi hi cos cp Present paper y/b b ; (02) For brevity the following notation is also adopted tan cp = 9 NACA ARR No. L4L14 17 3y us'^ of equations (C2) , the expres-sions from reference 1 that are used in the present paper may be rewritten as follows: N^,,iT^be Tc = w. ^xy 2\ f- (03) Vi = w, 2 IZR ^ 4b\ (^J(l.e2ff,-:-.(l.5e2)fo.(|)'f, (C4) 2 Tr^L\e 2lD^' (C5) v>/here / rr£_ \ 120 + G - 2_ Tr2y e + f2 24" 2\ 4\ ^. = (1-;^^^?-^-^ and c is the restraint coef f it-.lent defined in appendix B. The work done by the compressive force per half wave length may be written T — J- c b 2^'y / A, /(i\v'\2 dx dy :6) As in reference 1, the assumed deflection function is 18 NACA ^.RR No. L4L14 w rS-^V(-i) _c \^i3^ Try cos -^ T! . COS ^[x + 9y) (C7) When the e.'^presrion I'^or w from eiustion (C7) is substituted in (C6) and the indicated operations are performed, w. -ir-r^^2 + - fi (C8) Valur? from equations (C3) , (04), (C5), and (CS) are nonv substituted in the buckling equation (CI) . The use of the equations - V TT D = k 0^2 elirninatec N^y and Ny. The resulting equation gives the critical combination of stresses and may be written 9 '^ > ? 2f. \2 >2 (l + 9^)"fi + -pfg + 2(1 + 392)f2 tt2 b'' r^ ,S pf2 + 6 f T (C9) This equation shows tbat, for a selected value of k^, the critical shear stress depends upon the wave length and the angle of the buckle. Since a structure buckles at the lowest stres^. at which instability can occur, kg is T.inimized with respect to wave length and angle NACA ARR No. L4L14 19 of buckle. The inlnlnum value of kg with respect to value of wave length is determined from the condition 6kg = (GIO) which gives (when € does not depend on wave length) a) = 2 (l + 6^)\/fi |f 3 - % - kef 2 (Cll) Substitution of this value of wave length in equa- tion (C9) gives ks = 1 + 3^ j r flB 1^ fl ^ + f^ ) - k„f2 lA > + -^|2(l + 30^)f2 - kc^^^lj ^^12^ The m.inimum value of kg with respect to angle of buckle is found from the condition -69 = (C13) which gives fi('^ + f-. ^0^2 1/ c. + fc -C ^ll^ ^ ^^ ^0^2 1/2 + 2fp kcf^l (C14) N«CA ARR No. L4L14 Tf this value of 9 if substituted in (012), the final result is the following interaction formula, in which kg is elven in terms of k^ and the edge restraint e: ;(C2 - ^-c)-" (4^2 - k^)xj2(?.Ci - Cgke) (C15) where f, + H ^1 " h /l 1 ^.2 , /I 2 V + 1 c -. ^ and C.- = — ^ fn (A - 4)-^ - (f - A)- 1 NACii «RR No. L4L14 21 REPSREPCSS 1. Stowell, Elbridge Z., and. Scliv/arts, E.d'.vard 3.: Critical Stress for an InriniteTy Long Flat Plate with Elastically Restrained Edges under Combined Shear and Direct Stress. I'ACA' ARR No. 3K13, 1943, 2. Shanley, F. R., and Ryder, S. I.s Stres? Ratios. The Answer to the Combined Loading Problem. Aviation, vol. 3c, no. 6, June 1937, pp. 28, 29, 3. Anon.: Strength of Aircraft Elements. ANC-5, Army- Navy-Civil Coimnittee on Aircraft Design Criteria, Revised ed., Dec. 1942. 4. Crate, Harold, Batdorf, S. B., and Baab, George W. ; The Effect of Internal Pressure on the Buckling Stress of Thin-''rtalled Circular Cylinders vmder Torsion. NACA ARR No. L4E27, 194 4. 5. Hopkins, H. G., and Rao, B. V. 3. C: The Initial Buckling of Flat Rectangular Panels under Combined Shear and Compression. Rep. No. S.M.E. 3244, British R.A.S., ?'arch 1943. 6. Wagner, Herbert: IJber Plonstruktions- und Eerechnungsf ragen des Blechbaues. Jahrb . WGL, 1928, R. Cldenbourg (Kiinchen -xad Berlin), pp. 113- 125. 7. Timoshenko, S.: Theory of Elastic Stability. McGraw- Hill Book Co., Inc., 1936, p. 305. 8. Stowell, Elbridge Z.: Critical Shear Stress of an Infinitely Long Flat Plate with Equal Elastic Restraints against Rotation along the Parallel Edges. NACA ARE No. 3K12, 1943. 9. Lundqulst, Eugene E. , Rossm.an, Car] A., and Houbolt, John C: A '^lethod for Determining the Column Curve from Tests of Columns with Equal Restraints against Rotation on the Ends. NACA TN No. 903, 1943. NACA ARF No. L4L14 22 TABLE I.- VALUES OP k^ AND R^ COMPUTED VALUES ^'^ k. WITH CORRESPONDING AND R, ^C k. Ra Exact Energy Re Exact ( - 1.00 ( ° I- 2.86 3.16 } 1.00 \ I .535 .99 3.09 3.39 .99 .578 .95 3.36 3.67 .95 .629 .90 3.57 3.88 .90 .669 .80 3.87 4.20 .80 .725 .60 4.35 4.66 .60 .814 .40 4.70 5.04 .40 .881 .30 4.88 5.20 .30 .913 .20 5.03 5.36 .20 .942 .10 5.19 5.51 .10 .972 5.34 5.66 1.000 -1.00 6.56 6.89 -1.00 1.229 -2.00 7.63 7.91 -2.00 1.428 -5,00 10.22 10.47 -5.00 1.914 € = 2 1.67 f ° I 2.77 2.96 ) 1.00 ( ° I .457 1.32 4.07 4.32 .80 .672 .67 5.20 5.44 .40 .858 6.06 6.30 1.000 -3.34 9.26 9.50 -2.00 1.527 -8.34 12.95 13.16 -5.00 2.137 t = 5 2.30 f I 2.76 3.03 j 1.00 ^ .413 1.84 4.27 4.43 .80 .639 .92 5.62 5.81 .40 .841 6.68 6.88 1.000 -4.59 10.67 10.88 -2.00 1.597 -11.49 15.28 15.52 -5.00 2.267 < = 10 2.85 ' 2.78 3.18 1.00 f >• .382 2.28 4.49 4.60 .80 .618 1.14 6.04 6.19 .40 .831 7.27 7.43 1.000 -5.71 11.87 12.13 -2.00 1.633 -14.27 17.29 17.60 -5.00 2.378 t = CO 4.00 ' 3.27 3.81 ] 1.00 f ° I- .364 3.80 4.27 4.34 .95 .476 3.60 4.73 4.76 .90 .527 3.20 5.42 5.45 .80 .604 2.40 6.52 6.57 .60 .726 1.60 7.42 7.52 .40 .826 .80 8.24 8.37 .20 .917 8.98 9.15 1.000 -4.00 12.21 12.52 -1.00 1.360 -8.00 14.95 15.39 -2.00 1.666 -20.00 21.90 82.67 -5.00 2.439 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS NACA ARR No. L4L14 Fig. / ^ ORY INAUTICS -wg / _i o / NATION IMinEE -/ / o <-> A'/ r // o // / / / i // /// / '/// ►— / f/ ] / / '/ 1 / / k-1 / //// / 1 /// * 1 ' ll // 1 1 / 7 «0 ^4 tfl £ c °^" (U 0) in -*- o o in 1 0) -♦- Q.*- r o in U1 <0 Oi E 1 o 1L_ 13 C O CP — o (/) ft> >>_ c «A o CJ o > ^"n c o h -♦- C *) o o ^ o ■n c h >» fT) "o o o _ o . o «rt O o I 01: I 4) *♦- 3 O -f- «n C O QJ ft) O T3 lO 5t 1/) c T) C .^ -"~ o c o ■o c I. a m u iO I. 0) (rt ii. a: >^ Q. E o o o •4- u u o •+- i. 1/) o o in v k. «/> c rr^ 0) o 1 CT in »_ T5 c O (U «> 1) ^ -t- £i U 1/) tr ^ # +- «f- 5 O 0) +- ;n o o C o a Q) -t- en _c o c tn c o S E o o 'c o -r^ o .«_ r i. o u 1 • o rO GH 4J O \> ■+- 3 o D1 ^ NACA ARR No. L4L14 Fig. 3b .0 .6 ,6 R. .4 .2 ■ UUUU^-^I t-f-t-t-t-f-t-t--! NATIONAL ADVISORY C|)MiyHTTE£ FOR AERONAUTICS .4 .6 .6 (b) Shear with compression. Figure 3, ~ Concluded. NACA ARR No. L4L14 Fig. 4 R, .4 .6 .6 1.0 p NATIONAL ADVISORY ^ COMMIHEE FOR AERONAUTICS Figure 4.- Comparison of correct in+eraction curves with a curve formerly proposed for infinitely 'ong plates under combined shear and compression . NACA ARR No. L4L14 Figs. 5,6 i-^i-^iJli-^Li^l-^1 -^ I- T^T^t^t^t^TT'T'^t' y — ^v— ^i — ^i Jb 2 2 f X l^t-t-! Figure 5, - Coordinate system used in Appendix B for on infinitely long plate under connbined shear and transverse compression. UL^iia^^ij,^4_l_jj^ r^t^I^T^T'^in^T^I^T^I^T NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS Figure 6. - Coordinate system used and wave form assumed for energy solution in Appendix Cj inclined lines indicate nodal lines of buckles. UNIVEHSITY OF FLORIDA 3 1262 08106 513 7 UNIVERSIPj^ of FLORiDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE LIBRARY RO. BOX 117011 GAINESVILLE, FL 32611-7011 USA \